Calculating weights is a common analysis method. In actual research, it needs to be selected based on the characteristics of the data. For example, the volatility between data is an amount of information, then you can consider using the CRITIC weighting method or the information weighting method; Or maybe the expert scores the data, then AHP hierarchical method or prioritized graph method can be used.
This article lists common weight calculation methods, and compares the ideas and general principles of various weight calculation methods, conditions of use, etc., so that researchers can choose scientific weight calculation methods.
First, the common 8 types of weight calculation methods are listed, as shown in the following table:
The calculation principles of these 8 types of weights are different. The principles of calculating weights by combining various methods can be roughly divided into 4 types, as follows:
- The first category is factor analysis and principal component method; this method uses the information enrichment principle of data, and uses the variance interpretation rate to calculate the weight;
- The second category is AHP hierarchical method and optimal order graph method; this method uses the relative size information of numbers to calculate the weight;
- The third type is the entropy method (entropy weight method); this method uses the data entropy value information, that is, the amount of information to calculate the weight;
- The fourth category is CRITIC, independence weight and information weight; this kind of method mainly uses the volatility of the data or the correlation between the data to calculate the weight.
The first category, information enrichment (factor analysis and principal component analysis)
When calculating the weights, both the factor analysis method and the principal component method can calculate the weights, and the principle of use is exactly the same, all using the idea of information concentration. The difference between the factor analysis method and the principal component method is that the factor analysis method has a function of 'rotation', and the purpose of the principal component method is to concentrate information. The 'rotation' function can make the factor more interpretable. If you want the extracted factor to be interpretable, you usually use factor analysis method; it is not that the result of the principal component is completely unexplainable, but sometimes its interpretation The performance is relatively poor, but its calculation is faster, so it is widely used.
For example, there are 14 analysis items. The 14 items can be condensed into 4 aspects (also called factors or principal components). At this time, what are the weights of the 4 aspects? This is the principle of factor analysis or principal component calculation of weights. It uses the principle of information extraction to condense 14 items into 4 aspects (factors or principal components). The amount of information extracted by each factor or principal component (variance Interpretation rate) can be used to calculate the weight. Next, take SPSSAU as an example to explain the specific use of factor analysis to calculate the weight.
If it is expected that the 14 items can be divided into 4 factors, then the 4 factors can be actively set to extract, which is equivalent to 14 sentences can be concentrated into 4 keywords.
But sometimes it is not known how many factors should be more suitable. At this time, it can be combined with the results of the automatic recommendation of the software and professional knowledge to make a judgment. After clicking SPSSAU 'Start Analysis', the key table output is as follows:
The yellow shading in the above table is the 'rotation forward difference interpretation rate', which is the result before no rotation, which is essentially the result of the principal component. If factor analysis is used, the result corresponding to the 'rotational posterior difference interpretation rate' is generally used.
The variance explanation rate% in the results indicates the amount of information extracted by each factor. For example, the amount of information extracted by the first factor is 22.3%, the second factor is 21.862%, the third factor is 18.051%, and the fourth factor is 10.931% . And the amount of information extracted by the four factors is 73.145%.
Then the current 4 factors can express 14 items, and the accumulated information amount of 14 factors extracted by the 4 factors is 73.145%. Now we hope to get the weights of the four factors. At this time, we can use the normalization process, which means that the four factors all represent the entire 14 items. Then the information of the first factor is 22.3% / 73.145% = 30.49%; The similar second factor is 21.862% / 73.145% = 29.89%; the third factor is 18.051% / 73.145% = 24.68%; the fourth factor is 10.931% / 73.145% = 14.94%.
If the principle component method is used for weight calculation, the principle is similar. In fact, the result is the corresponding calculation of the value of the 'rotation forward difference interpretation rate' value.
When using the principle of condensed information for weight calculation, only the weight of each factor can be obtained, and the weight of each analysis item cannot be obtained. At this time, it is possible to continue to combine the subsequent weight method (usually the entropy method) to obtain the specific items. The weights are then aggregated together to finally build a weight system.
The core point of weight calculation through factor analysis or principal component analysis is to obtain the variance interpretation rate value, but before getting the weight, in fact, there is still more preparation work, such as extracting 4 factors in this example, why is 4 Not 5 or 6; this is judged by combining other indicators extracted by professional knowledge and analysis methods; and sometimes some analysis items are not suitable for analysis, and they need to be deleted before they can be analyzed. The preparation work is prepared before the analysis. For details, please refer to the SPSSAU help manual for specific practical cases and video descriptions.
The second category, the relative size of numbers (AHP hierarchical method and optimal order graph method)
计算权重的第二类方法原理是利用数字相对大小，数字越大其权重会相对越高。此类原理的代表性方法为AHP层次法和优序图法。
1. AHP层次法
AHP层次分析法的第一步是构建判断矩阵，即建立一个表格，表格里面表述了分析项的相对重要性大小。比如选择旅游景点时共有4个考虑因素，分别是景色，门票，交通和拥护度，那么此4个因素的相对重要性构建出判断矩阵如下表：
表格中数字代表相对重要的大小，比如门票和景色的数字为3分，其说明门票相对于景色来讲，门票更加重要。当然反过来，景色相对于门票就更不重要，因此得分为1/3=0.3333分。
AHP层次分析法正是利用了数字大小的相对性，数字越大越重要权重会越高的原理，最终计算得到每个因素的重要性。AHP层次分析法一般用于专家打分，直接让多位专家（一般是4~7个）提供相对重要性的打分判断矩阵，然后进行汇总（一般是去掉最大值和最小值，然后计算平均值得到最终的判断矩阵，最终计算得到各因素的权重。
SPSSAU共有两个按键可进行AHP层次分析法计算。
如果是问卷数据，比如本例中共有4个因素，问卷中可以直接问“景色的重要性多大？”，“门票的重要性多大？”，“交通的重要性多大？”，“拥护度的重要性多大？”。可使用SPSSAU【问卷研究】--【权重】，系统会自动计算平均值，然后直接利用平均值大小相除得到相对重要性大小，即自动计算得到判断矩阵而不需要研究人员手工输入。
AHP层次分析：【问卷研究】--【权重】
如果是使用【综合评价】--【AHP层次分析法】，研究人员需要自己手工输入判断矩阵。
【综合评价】--【AHP层次分析】
2. 优序图法
除了AHP层次分析法外，优序图法也是利用数字的相对大小进行权重计算。
数字相对更大时编码为1，数字完全相同为0.5，数字相对更小编码为0。然后利用求和且归一化的方法计算得到权重。比如当前有9个指标，而且都有9个指标的平均值，9个指标两两之间的相对大小可以进行对比，并且SPSSAU会自动建立优序图权重计算表并且计算权重，如下表格：
优序图法
上表格中数字0表示相对不重要，数字1表示相对更重要，数字0.5表示一样重要。
比如指标2的平均值为3.967，指标1的平均值是4.1，因此指标1不如指标2重要；指标4的平均值为4.3，重要性高于指标1。也或者指标7和指标9的平均得发均为4.133分，因此它们的重要性一样，记为0.5。
结合上面最关键的优序图权重计算表，然后得到各个具体指标（因素）的权重值。
优序图法适用于专家打分法，专家只需要对每个指标的重要性打分即可，然后让软件SPSSAU直接结合重要性打分值计算出相对重要性指标表格，最终计算得到权重。
优序图法和AHP法的思想上基本一致，均是利用了数字的相对重要性大小计算。一般在问卷研究和专家打分时，使用AHP层次分析法或优序图法较多。
第三类、信息量(熵值法)
计算权重可以利用信息浓缩，也可利用数字相对重要性大小，除此之外，还可利用信息量的多少，即数据携带的信息量大小（物理学上的熵值原理）进行权重计算。
熵值是不确定性的一种度量。信息量越大，不确定性就越小，熵也就越小；信息量越小，不确定性越大，熵也越大。因而利用熵值携带的信息进行权重计算，结合各项指标的变异程度，利用信息熵这个工具，计算出各项指标的权重，为多指标综合评价提供依据。
在实际研究中，通常情况下是先进行信息浓缩法（因子或主成分法）得到因子或主成分的权重，即得到高维度的权重，然后想得到具体每项的权重时，可使用熵值法进行计算。
SPSSAU在【综合评价】模块中提供此方法，其计算也较为简单易懂，直接把分析项放在框中即可得到具体的权重值。
【综合评价】--【熵值法】
熵值法：SPSSAU熵值法帮助手册
第四类、数据波动性或相关性(CRITIC、独立性和信息量权重)
可利用因子或主成分法对信息进行浓缩，也可以利用数字相对大小进行AHP或优序图法分析得到权重，还可利用物理学上的熵值原理（即信息量携带多少）的方法得到权重。除此之外，数据之间的波动性大小也是一种信息，也或者数据之间的相关关系大小，也是一种信息，可利用数据波动性大小或数据相关关系大小计算权重。
1. CRITIC权重法
CRITIC权重法是一种客观赋权法。其思想在于用两项指标，分别是对比强度和冲突性指标。对比强度使用标准差进行表示，如果数据标准差越大说明波动越大，权重会越高；冲突性使用相关系数进行表示，如果指标之间的相关系数值越大，说明冲突性越小，那么其权重也就越低。权重计算时，对比强度与冲突性指标相乘，并且进行归一化处理，即得到最终的权重。使用SPSSAU时，自动会建立对比强度和冲突性指标，并且计算得到权重值。
CRITIC权重法适用于这样一类数据，即数据稳定性可视作一种信息，并且分析的指标或因素之间有着一定的关联关系时。比如医院里面的指标：出院人数、入出院诊断符合率、治疗有效率、平均床位使用率、病床周转次数共5个指标；此5个指标的稳定性是一种信息，而且此5个指标之间本身就可能有着相关性。因此CRITIC权重法刚好利用数据的波动性（对比强度）和相关性（冲突性）进行权重计算。
SPSSAU综合评价里面提供CRITIC权重法，如下图所示：
【综合评价】--【CRITIC权重】
2. 独立性权重法
独立性权重法是一种客观赋权法。其思想在于利用指标之间的共线性强弱来确定权重。如果说某指标与其它指标的相关性很强，说明信息有着较大的重叠，意味着该指标的权重会比较低，反之如果说某指标与其它指标的相关性较弱，那么说明该指标携带的信息量较大，该指标应该赋予更高的权重。
独立性权重法仅仅只考虑了数据之间相关性，其计算方式是使用回归分析得到的复相关系数R 值来表示共线性强弱（即相关性强弱），该值越大说明共线性越强，权重会越低。比如有5个指标，那么指标1作为因变量，其余4个指标作为自变量进行回归分析，就会得到复相关系数R 值，余下4个指标重复进行即可。计算权重时，首先得到复相关系数R 值的倒数即1/R ，然后将值进行归一化即得到权重。
比如某企业计划招聘5名研究岗位人员，应聘人员共有30名，企业进行了五门专业方面的笔试，并且记录下30名应聘者的成绩。由于专业课成绩具有信息重叠，因此不能简单的直接把成绩加和用于评价应聘者的专业素质。因此使用独立性权重进行计算，便于得到更加科学客观的评价，选出最适合的应聘者。
SPSSAU综合评价里面提供独立性权重法，如下图所示：
【综合评价】--【独立性权重】
3. 信息量权重法
信息量权重法也称变异系数法，信息量权重法是一种客观赋权法。其思想在于利用数据的变异系数进行权重赋值，如果变异系数越大，说明其携带的信息越大，因而权重也会越大，此种方法适用于专家打分、或者面试官进行面试打分时对评价对象（面试者）进行综合评价。
比如有5个水平差不多的面试官对10个面试者进行打分，如果说某个面试官对面试者打分数据变异系数值较小，说明该面试官对所有面试者的评价都基本一致，因而其携带信息较小，权重也会较低；反之如果某个面试官对面试者打分数据变异系数值较大，说明该面试官对所有面试者的评价差异较大，因而其携带信息大，权重也会较高。
SPSSAU综合评价里面提供信息量权重法，如下图所示：
【综合评价】--【信息量权重】
对应方法的案例说明、结果解读这里不再一一详述，有兴趣可以参考SPSSAU帮助手册。
CRITIC权重：SPSSAU-CRITIC权重帮助手册
独立性权重：SPSSAU-独立性权重帮助手册
Information weight: SPSSAU-information weight help manual